极化流形的拟射模(Eckart Viehweg)Quasi projective Moduli for Polarized Manifolds (Eckart

本书讨论了两个性质截然不同的主题：通过群作用或等价关系和光滑的射影下某些滑轮的直接像的性质来构造拟投影方案的商。PrefaceThe concept of moduli goes back to B Riemann, who shows in 68 that theisomorphism class of a Riemann surface of genus g >2 depends on 39-3parameters, which he proposes to name "moduli. A precise formulation ofglobal moduli problems in algebraic geometry, the definition of moduli schemesor of algebraic moduli spaces for curves and for certain higher dimensionalmanifolds have only been given recently(A. Grothendieck, D. Mumford, see59), as well as solutions in some casesIt is the aim of this monograph to present methods which allow over afield of characteristic zero to construct certain moduli schemes together withan ample sheaf. Our main source of inspiration is D. Mumfords "Geometric Invariant Theory". We will recall the necessary tools from his book [59 and provethe " Hilbert-Mumford Criterion" and some modified version for the stabilitvof points under group actions. As in 78, a careful study of positivity properties of direct image sheaves allows to use this criterion to construct moduli asquasi-projective schemes for canonically polarized manifolds and for polarizedmanifolds with a semi-ample canonical sheafFor these manifolds moduli spaces have been obt ained beforehand as ana-lytic or algebraic spaces(631,74,41, 66,59, Appendix to Chapter 5, and44. We will sketch the construction of quotients in the category of algebraicspaces and of algebraic moduli spaces over an algebraically closed field k of anycharacteristic, essentially due to M. Artin. Before doing so, we recall C. s Sehadri's approach towards the construction of the normalization of geometricquotients in[ 71. Using an ampleness criterion, close in spirit to stability criteria, in geometric invariant theory, and using the positivity properties mentionedabove, his construction will allow to obtain the normalization of moduli spacesover a field of characteristic zero as quasi-projective schemes. Thereby the al-gebraic moduli spaces turn out to be quasi-projective schemes, at least if theyare normal outside of a proper subspaceFor proper algebraic moduli spaces, as J. Kollar realized in 47, it is sufficient to verify the positivity for direct images sheaves over non-singular curvesThis approach works as well in characteristic p>0. llowever, the only moduliproblem of polarized manifolds in characteristic p>>0, to which it applies atpresent, is the one of stable curves, treated by F. Knudsen and D. Mumford bydifferent methodsriAceCompared with 78,79 and [18 the reader will find simplified proofs, butonly few new results. The stability criteria are worked out in larger generalityand with weaker assumptions than in loc cit. This enables us to avoid the cumbersome reference in the positivity results to compactifications, to enlarge theset of ample sheaves on the moduli schemes and to extend the methods of construction to moduli problems of normal varieties with canonical singularities,provided they are "locally closed and bounded. Writing this monograph werealized that some of the methods, we and others were using, are well-knownto specialists but not documented in the necessary generality in the literatureWe tried to include those anld most of the results which are niot contained instandard textbooks on algebraic geometry, with three exceptions: We do notpresent a proof of "Matsusaka,'s Big Theorem", nor of Hilberts theorem onrings of invariants under the action of the special linear group. in spite of theirimportance for the construction of moduli schemes. And we just quote the results needed froin the theory of canonical singularities and canonical modelswhen we discuss moduli of singular schemesNevertheless, large parts of this book are borrowed from the work ofothers, in particular fron D. Muinford's book [59, C.S. Seshadri's article71,J. Kollar's articles 44 and [47, from 501, written by J Kollar andN. I. Shepherd-Barron, fron [18 and the Lecture Notes[19, both written withH. Esnault as coauthor. Besides, our presentation was partly influenced by theLecture Notes of D. Gieseker 261, D. KIutson [43, P. E Newstead [64] andH.POpp[66」As to acknowledgements I certainly have to mention the " Max-Planck-Institut fur Mathematik", Bonn, where I started to work on Inoduli probleMsduring the"Special year on algebraic geometry(1987/88) " and the "IHESBures sur Yvette, where the second and third part of 78 was finished. Dur-ing the preparation of the manuscript I was supported by the dfG(German Research Council) as a menber of the " Schwerpunkt KoInplexe mannifaltigkeitell and of the "Forschergruppe Arithinetik und geometrieI owe thanks to several mathematicians who helped me during different periods of my work on moduli schemes and during the preparation ofthe manuscript, among them R. Hain, E. Kani, Y. Kawamata, KollarN. Nakayama, V. Popov and C. S. Seshadri. Without O. Gabber, telling meabout his extellsioIl theoren and its proof, presumably I would not have beenable to obtain the results on the positivity of direct image sheaves in the gen-erality needed for the construction of moduli schemes. G. Faltings, S. KeelJ Kollar, L Moret-Bailly and L Ramero pointed out mistakes and ambiguitiesin an earlier version of the manuscriptThe infuence of Helene Esnault on the content and presentation of thisbook is considerable. She helped me to clarify several constructions, suggestedimprovements, and part of the methods presented here are due to her or to ourcommon workEssen. March 1995Eckart ViehwegTable of contentsIntroductionLeitfadenClassification Theory and Moduli ProblemsNotations and Conventions121 Moduli problems and hilbert schemes1.1 Moduli functors and moduli schemes1.2 Moduli of manifolds: The main results1.3 Properties of Moduli Functors221.4 Moduli Functors for Q-Gorenstein Schemes271.5 A. Grothendiecks Construction of Ililbert Schemes301.6 Hilbert Schemes of Canonically Polarized Schemes431.7 hilbert Schemes of polarized schemes472 Weakly positive sheaves and vanishing theorems32.1 Coverings532.2 Numerically Effective Sheaves572.3 Weakly Positive sheaves2.4 Vanishing Theorems and Base Change672.5 Examples of Weakly positive Sheaves3 D. Mumford's Geometric Invariant Theory3. 1 Group Actions and Quotients773.2 Linearizations3.3 Stable points913.4 Properties of Stable Points963.5 Quotients, without Stability Criteria4 Stability and Ampleness Criteria4.1 Compactifications and the Ililbert-Mumford Criterion1124.2 Weak Positivity of Line Bundles and Stability1194.3 Weak Positivity of Vector Bundles and Stability1274.4 Ampleness Criteria133Tablc of contents5 Auxiliary results on Locally Free Sheaves and Divisors1395.1 Gabber's extension theorem1405.2 The Construction of Coverings14855 Vanishing Theorems and Base Change. Revisite···1535.3 Singularities of divisors5.4 Singularities of Divisors in Flat Families1581646 Weak positivity of Direct Images of Sheaves1676.1 Variation of Hodge structures1686.2 Weakly Semistable reduction1703 Applications of the Extension Theorem,.1786.4 Powers of dualizing Sheaves1906.5 Polarizations, Twisted by Powers of Dualizing Sheaves1927 Geometric Invariant Theory on Hilbert Schemes1977. 1 Group Actions on Hilbert Schemes1987.2 Geometric Quotients and Moduli Schemes2057.3 Methods to Construct Quasi-Projective Moduli scheme207.4 Conditions for the Existence of Moduli Schemes: Case(CP2167.5 Conditions for the Existence of Moduli Schemes: Case (DP)2207.6 Numerical Equdivalence2248 Allowing Certain Singularities8.1 Canonical and Log-Terminal singularities,.,,,,2408.2 Singularities of divisors2438.3 Deformations of Canonical and Log-Terminal Singularities2478.4 Base Change and positivity2518.5 Moduli of Canonically polarized varieties,,2548.6 Moduli of polarized varieties2588.7 Towards Moduli of canonically Polarized Schemes2629 Moduli as algebraic Spaces2779.1 Algebraic spaces2789.2 Quotients by Equivalence relations9.3 Quotients in the Category of Algebraic Spaces2879. 4 Construction of Algebraic Moduli spaces909.5 Ample Line Bundles on Algebraic Moduli spaces2989.6 Proper Algebraic Moduli Spaces for Curves and Surfaces305References311Glossary of notations315Index317IntroductionB Riemann 68 showed that the conformal structure of a Riemann surface ofgenus 9>l is determined by 3g-3 paraneters, which he proposed to namemoduli".Following A. Grothendieck and D Mumford 59 we will consideralgebraic Inoduli' in this monograph. To give a flavor of the results we areinterested in, let us recall D. Munford's strengthening of B. Riemann's statementTheorem(Mumford [59) Let ki be an algebraically closed field and, forg> 2,(h)=projective curves of genus g, defined over h 1/iIsomorphismsThen there exists a guasi-projective coarse moduli variety Ca of dimension 3g-3i.e. a quasi-projective variety Ca and a natural bijection ea(k)= Co(k) whereC,(k denotes the k-valued points of CpOf course, this theorem makes sense only when we give the definition ofnatural"(see 1.10). Let us just remark at this point that "natural impliesthat for a fat morphism f: X-Y of schemes, whose fibers f(y belongto eo(k), the induced map Y(k)- Cg(h) should come from a morphism ofschemesIn the spirit of B. Riemann's result one should ask for a description ofalgebraic parameters or at least for a description of an ample sheaf on Cg. Vwill see in 7.9 that for each i>0 there is some p >0 and an invertible sheafA(p)on C. withP"Au)=(det(f wx/Y))pwhere o is the natural morphism from y to Cg. D. Mumford's construction ofimpliesAddendum(Mumford [59])For v, u and p sufFiciently large, Jo=(2g-2)·-(g-1)=(2g-2)·v·1-(9-1)the sheaf Ap)& Ap- is ampleTrying to generalize Mumfords result to higher dimensions, one first remarks that the genus g of a projective curve I determines the Hilbert polynomial h(T) of I. If wr denotes the canonical sheaf thenIntroductionh()=xn)=(29-2)·-(9-1)Hence,ifhT)∈Q门 is a polynomial of degree n, with h()∈ Z for v∈z,one should considerCh(k)=ir: I projective manifold defined over k, wr ampleand h(v)=xwr) for all vy/isomorphismsBince wir is ample for I E Ch(k), one has n= dim(I). If n=2, i.e. in thecase of surfaces, we will replace the word"manifold"in the definition of eh(kDy " normal irreducible variety with rational double points". Let us write eh(kfor this larger set. D. Gieseker proved the existence of quasi-projective modulischemes for surfaces of general typeTheorem(Gieseker [25) If char(k)=0 and deg(h)=2 then there exists aquasi-projective coarse moduli scheme Ch for C'hIf Ap) denotes the sheaf whose pullback to y is isomorphic to det(faw'xr)P,forall families f: X-Y of varieties in Ch(k), then app)' Ap)-l'l is ampleon Ch: for v and u sufficiently largeThe construction of moduli schemes for curves and surfaces of general typeuses geometric invariant theory, in particular the Hlilbert-Mumford Criterionfor stability(59,25 and 26). We will formulate this criterion and sketch itsproof in 4.10. Applied to points of llilbert schemes this criterion reduces theconstruction of moduli schemes to the verification of a certain property of themultiplication mapsS(H(F,1)一→H(F,y)for u>v> I and for all I in Ci(k) or in Ch(k). This property, formulatedand discussed in the first part of Section 7.3, has been verified for curves in 59(see also26) and for surfaces in 25]. For n >2 the corresponding property ofthe multiplication map is not known. In this book we will present a differentapproach which replaces the study of the multiplication map for the manifoldsTECn(k) by the study of positivity properties of the sheaves f*wx/Y for familiesf:X→YofThese positivity properties will allow to modify the approach used by mum-ford and Gieseker and to prove the existence of coarse quasi-projective modulischemes Ch for manifolds of any dimension, i. e. for deg(h)E N arbitrary. pro-vided char(k)=0. Unfortunately similar results over a field k of characteristicp>0 are only known for moduli of curves. The sheaves Ap)will turn out to beample on Ch for v sufficiently large. As we will see later( see 7. 14)for m <2 theample sheaves obtained by Mumford and Gieseker are"better"than the onesobtained by our methodLet us return to d. mumford's construction of moduli of curves. The moduli schemes Co have natural compactifications, i.e. compactifications which areIntroductithemselves moduli schemes for a set of curves containing singular ones. follow-ing A Mayer and D. Mumford, one defines a stable curve r of genus g> 2 as aconnected reduced and proper scheme of dimension one, with at most ordinarydouble points as singularities and with an ample canonical sheaf wr. The genusg is given by the dimension of H (T, wr). One hasTheorem(Knudsen, Mumford [42, Mumford 62) Let h be an, algebraicolly closed field and for g> 2Cg(ki)=stable curves of genus g, defined over k)/isomorphismsThen there erists a projective coarse moduli variety Cg of dimension 3g-3If ap) denotes the sheaf whose pullback to y is isomorphic to det(fwx/y)p, forall families f: X-Y of schemes in Ca, then入P)(-1)(2…-1⑧入(p)-9-1)(2-1is ample on Cg for v and u suficiently largeJ. Kollar and N. I. Shepherd-Barron define in 50 a class of reduced twodimensional schemes, called stable surfaces, which give in a similar way a coinpletion Ch of the noduli problen Ch of surfaces of general type QUite recentlyJ. Kollar [47] and V. Alexeev [1 finished the proof, that the corresponding moduli scheme exists as a projective scheine. In the higher dimensional cawill discuss at the end of this monograph, things look desperate. If one restrictsoneself to moduli problens of nornal varieties, one should allow varieties withcanonical singularities, but one does not know whether small deformations ofnese varieties have again canonical singularities. Apart from this, nost of ourconstructions go through. For reducible schenes we will list the properties areasonable completion of the moduli functor Ch should have, and we indicatehow to use the construction methods for moduli in this caseIn order to obtain moduli for larger classes of higher dimensional manifoldsone has to consider polarized manifolds, i.e. pairs(I, 7) where H is an ampleinvertible sheaf on I(see 59, p: 97). We define(H)≡(r,if there exists an isomorphism T: I+such that H and TH are numericallycquivalent. and(,7)~(,7)if there are isomorphisms T:T→r"andr*h→hIf h(r, f)=0, both equivalence relations are the same (up to torsionand both can be used to describe a theorem, which I.I. Piatetskii-Sapiro andI.R. Safarevich obtained by studying period mapsIntroductionTheorem(Pjatetskij-Sapiro and Safarevich [65 If h is a polynomial ofdegree 2, there exists a coarse quasi-projective modali scheme M forI(T, H); I a compler K-3 surface, H ample invertible on Tand h(v)=x(H"), for allv/=On M there is an ample invertible sheaf A(p) whose pullback to the base y of afamily f: X-Y K-8 surfaces is the sheaf (wx/y)pIf one considers d. mumfords theorem on moduli of abelian varieties. onefinds a third equivalence relation(,H)≡a(,%) if there are a,b∈N-0,with(,0)≡(",7")This relation occurs in a natural way since Mumford considers instead of Ha. morphism 4(H) from the abelian manifold I to its dual I. The morphismA(H) only depends on the numerical equivalence class of H. Only some powerof H can be reconstructed from 1(7). However, for moduli schemes of abelianvarieties it is not difficult to pass from“=”to"≡o”andD. Mumford's theorencan be restated asTheorem(Mumford 59) For h(r)E Qf there exists a coarse quasiprojective moduli scheme M forI(T, e, 7);I an abelian variety with unit element e, H ampleinvertible on I and h(v)=x(H)for allv/=As for K-3 surfaces, there is an ample invertible sheaf X(p)on M whosepullback to the base y of a family f: X-Y of abelian varieties is the sheaf(/yIt is unlikely that the last two theorems can be generalized to arbitrarymanifolds I. One has to exclude uniruled manifolds and manifolds with exceptional divisors. Hence it is natural to require that wr is numerically effectivewill allow to replace a given polarization by one which is "close w hs mptionor, in other terms. thatof the ample cone. This assurSince it is not known whether the condition "numerically effective is alocally closed condition, we will replace it by the slightly stronger one, thatur is semi-ample. The second main result will be the construction of quasi-projective moduli, over a field k of characteristic zero, form,(k)=(r, h);Tmanifold definedsemI-ample,H ample invertible on I and h(u)=x(H,) for all v/s well as for m,(k)=,(k)/Moduli of vector bundles or of sheaves on a given manifold will not appearat all in this book. The analytic theory of moduli, or algebraic moduli spaceswill only play a role in Paragraph 9. We will not use the language ofstacks, although it is hidden in the proof of 9.16